Follow the Flow: sets, relations, and categories as special cases of functions with no domain Adonai S. Sant'Anna∗ Ot´avioBuenoy Marcio P. P. de Fran¸caz Draft 2.6 Disclaimer This is a report of an ongoing research project. This text is supposed to work as a simple reference for seminars to be delivered in Brazil. Updated versions will be available soon. Abstract We introduce, develop, and apply a new approach for dealing with the intuitive notion of function, called Flow Theory. Within our frame- work all functions are monadic and none of them has any domain. Sets, proper classes, categories, functors, and even relations are special cases of functions. In this sense, functions in Flow are not equivalent to func- tions in ZFC. Nevertheless, we prove both ZFC and Category Theory are naturally immersed within Flow. Besides, our framework provides major advantages as a language for axiomatization of standard mathematical and physical theories. Russell's paradox is avoided without any equiva- lent to the Separation Scheme. Hierarchies of sets are obtained without any equivalent to the Power Set Axiom. And a clear principle of duality emerges from Flow, in a way which was not anticipated neither by Cat- egory Theory nor by standard set theories. Besides, there seems to be within Flow an identification not only with the common practice of do- ing mathematics (which is usually quite different from the ways proposed by logicians), but even with the common practice of teaching this formal science. Key words: functions, set theory, category theory. ∗Department of Mathematics, Federal University of Paran´a.E-mail: [email protected]. yDepartment of Philosophy, University of Miami. zCoordination of Pedagogy, Federal University of Paran´a 1 Contents 1 Introduction 2 2 Flow theory 7 2.1 Functions . 8 2.2 Sets and Proper Classes . 32 3 ZFC is immersed in Flow 34 3.1 ZFC Axioms . 34 3.2 ZFC translation . 36 4 Category theory is immersed in Flow 39 4.1 Category axioms . 40 4.2 Every static category is a category . 41 4.3 Categories translation . 43 4.4 Functors and natural transformations . 45 4.5 Set and other standard categories . 46 4.6 The Cantor-Schr¨oder-Bernsteintheorem . 46 5 Intuitive Flow theory 47 6 Axiomatization as a flow-theoretic predicate 48 6.1 Group theory . 48 6.2 Other mathematical theories . 49 6.3 Classical particle mechanics . 49 6.4 Reformulating classical particle mechanics . 51 7 The full potential of Flow 52 7.1 Composition . 52 7.2 n-ary functions . 54 7.3 Mathematics teaching . 56 8 Variations of Flow 57 8.1 Closure . 57 8.2 Regularity . 58 8.3 Clones and equiconsistency . 58 9 Final remarks 59 10 Acknowledgements 59 1 Introduction Throughout the ages mathematicians have considered their objects, such as numbers, points, etc., as substantial things in themselves. 2 Since these entities had always defied attempts at an adequate de- scription, it slowly dawned on the mathematicians of the nineteenth century that the question of the meaning of these objects as substan- tial things does not make sense within mathematics, if at all. The only relevant assertions concerning them do not refer to substantial reality; they state only the interrelations between mathematically “undefined objects" and the rules governing operations with them. What points, lines, numbers \actually" are cannot and need not be discussed in mathematical science. What matters and what corre- sponds to \verifiable” fact is structure and relationship, that two points determine a line, that numbers combine according to certain rules to form other numbers, etc. A clear insight into the neces- sity of a dissubstantiation of elementary mathematical concepts has been one of the most important and fruitful results of the modern postulational development. Richard Courant, What is Mathematics, 1941. All usual mathematical approaches for well-known physical theories can be easily associated to either differential equations or systems of differential equa- tions. Newton's second law, Schr¨odinger'sequation, Maxwell's equations, and Einstein field equations are all differential equations which ground classical me- chanics, quantum mechanics, classical electromagnetism, and general relativity, respectively. Other similar examples may be found in thermodynamics, gauge theories, the Dirac electron, etc. Solutions for those differential equations (when they exist) are either functions or classes of functions. So, the concept of func- tion plays a major role in theoretical physics. Actually, functions are more relevant than sets, in a very precise sense [4] [5]. In pure mathematics the situation is no different. Continuous functions, linear transformations, homomorphisms, and homeomorphisms, for example, play a fundamental role in topology, linear algebra, group theory, and differential geometry, respectively. And category theory emphasizes such a role in a very clear, elegant, and comprehensive way. Functions allow us to talk about the dynamics of the world, in the case of physical theories. Regarding mathematics, functions allow us to talk about invariant properties, whether those properties refer to either algebraic operations or order relations. From a historical point of view, some authors have advocated the idea that functions are supposed to play a strategic role into the foundations of mathemat- ics [17] and even mathematics teaching [9], rather than sets. Notwithstanding, the irony of such discussions lies in a closer look at Georg Cantor's seminal works about the concept of set. Cantor - the celebrated father of set theory - was strongly motivated by Bernard Bolzano's work on infinite multitudes called Menge [25]. Those collections were supposed to be conceived in a way such that the arrangement of their components is unimportant. However, Bolzano insisted on an Euclidian view that the whole should be greater than a part, while Cantor proposed a quite different approach. According to the latter, in 3 order to compare infinite quantities we should consider a one-to-one correspon- dence between collections. That means Cantor's concept of collection (in his famous Mengenlehre) was strongly committed to the idea of function. Subse- quent formalizations of Cantor's \theory" were developed in a way such that all strategic terms were associated to an intended interpretation of collection. And the result of that effort is a strange phenomenon which we describe in the next paragraphs, based on [21]. Let S be an axiomatic system whose primitive concepts are c1, c2, ..., cn. One of these concepts, say ci, is independent (undefinable) from the remaining if and only if there are two models of S in which c1, ..., ci−1, ci+1, ..., cn have the same interpretation, but the interpretations of ci in such models are different. (Of course, a model of S is a set-theoretic structure in which all axioms of S are true, according to the interpretation of its primitive terms [15].) As an example, consider a very simple axiomatic system, namely, a Mini- malist Space hX; fi, whose axioms are: MS1 X is a non-empty set. MS2 f is a function whose domain and codomain are both X. By using Padoa's method [1] [18] [24] we can easily prove that f is undefin- able, since we can exhibit two models of a minimalist system such that X has the same interpretation in both models but f has two different interpretations within these models. Consider, for this: the Model A, where X is interpreted as the set of real numbers < and f is the identity function f(x) = x defined on X; and the Model B, where X is interpreted again as the set <, but f is the function given by f(x) = 2x, with the same domain X. This means that the interpretation of X does not fix the interpretation of f. In other words, f can- not be defined (or fixed) from X. On the other hand, X is definable, since any two models with two different interpretations for X would unavoidably entail different interpretations for f. The reason for this is grounded on the fact that the domain and the codomain of a function f are ingredients for the definition of the function itself, at least within the scope of a standard set theory like Zermelo-Fraenkel's. Different domains imply different functions. So, at least two questions remain: 1. How to define X? 2. What does it mean to say that X is eliminable? The answers are: 1. X = dom(f) = cod(f)(X is the domain and the codomain of f). 2. We do not need to explicitly mention X. We could rephrase the definition of a minimalist system by saying that a minimalist system is just a function f whose domain is equal to its codomain. 4 In a similar way, it is possible to prove that in usual axiomatic frameworks for physical theories, time and spacetime are concepts that are definable, and so, eliminable. That happens because time and spacetime are usually considered as domains of functions that describe forces, fields, currents, and so on. For example, according to Padoa's principle, the primitive concept time (described as an interval of real numbers) in a physical theory is independent from the remaining primitive concepts (mass, position, force, speed, magnetic field etc.) if, and only if, there are two models of the physical theory such that time has two interpretations and the remaining primitive symbols have the same inter- pretation. But usually these two interpretations are not possible, since mass, position, force, speed, magnetic field and other physical concepts are in general described as functions whose domains are time. If we change the interpreta- tion of time, we change the interpretation of the other primitive concepts. So, time is not independent and hence can be defined. Since time is definable, it is eliminable.